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132 | Journal of Digital Learning in Teacher Education | Volume 26 Number 4

Technological Pedagogical Content Knowledge in the Mathematics Classroom

Shannon GuerreroNorthern Arizona University

Abstract

Teacher knowledge has long been a focus of many educational research-ers. Current conceptualizations of teacher knowledge are beginning to reflect the knowledge and skills teach-ers need to successfully navigate in-creasingly technologically-rich math-ematical classrooms with the addition of knowledge domains such as techno-logical pedagogical content knowledge (TPACK). This article situates TPACK in the mathematics classroom by de-veloping four central components of knowledge necessary for technology-using mathematics teachers. This article concludes by presenting a por-trait of effective TPACK in action and posing questions for technology-using teachers to consider as they embark on technology use in support of math-ematics instruction. The intent of this article is not to offer a one-size-fits-all solution to the many issues surround-ing technology use, but to provide the impetus for discussion and reflection among mathematics educators at all levels. (Keywords: TPACK, teacher knowledge, technology, mathematics)

Research on teachers’ knowledge and beliefs has long remained a substantial area of inquiry in our

explorations of the nature of teaching. How teachers’ understanding of teach-ing, learning, students, and subject matter affects their everyday practice is an important aspect of our quest to understand the complex nature of teaching and the professional knowledge necessary for effective teaching. The influence of cognitive psychology on our understanding of learning has resulted in a number of ways to characterize the structure of knowledge in the individual (Borko & Putnam, 1996).

The development and analysis of teachers’ formal knowledge by various researchers and educators is an effort to explain various areas of expertise related to teaching and to develop rationale the-ories on which to base practice (Fried-son, 1986). Koehler and Mishra (2008) argue that teachers are autonomous agents with the power to significantly influence appropriate and inappropriate teaching. Thus, an understanding of the knowledge teachers must possess and access in various instructional settings has the potential to impact both teacher training and instructional practices.

With the emergence of technology’s integral role in our daily lives and edu-cational landscape at the beginning of the 21st century, researchers have begun to address the impact technology has on teacher knowledge. Although some researchers have begun by looking at the intersection of pedagogy and technol-ogy in the development of non-content-specific knowledge domains such as pedagogical technology knowledge (PTK) (Guerrero, 2005), others have examined the intersection of pedagogy and technology in the development of content-specific technological pedagogi-cal content knowledge (TPACK) (Koe-hler & Mishra, 2008; Koehler, Mishra, & Yahya, 2007; Pierson, 2001).

TPACKThe TPACK (formerly TPCK) frame-work expands on Shulman’s (1986a, 1986b, 1987) conceptualization of pedagogical content knowledge “…to describe how teachers’ understanding of technologies and pedagogical content knowledge interact with one another to produce effective teaching with technol-ogy” (Koehler & Mishra, 2008, p. 12). In this model (see Figure 1), peda-gogical knowledge, content knowledge,

and technology knowledge intersect, interact, and influence one another to form and inform not only a teacher’s understanding of content, pedagogy, and technology, but also combinations of these three knowledge domains. Together, these multiple knowledge do-mains intersect in the realm of TPACK to represent

…an understanding of the repre-sentation of concepts using tech-nologies; pedagogical techniques that use technologies in con-structive ways to teach content; knowledge of what makes con-cepts difficult or easy to learn and how technology can help redress some of the problems that stu-dents face; knowledge of students’ prior knowledge and theories of epistemology; and knowledge of how technologies can be used to build on existing knowledge and to develop new epistemologies or strengthen old ones. (Koehler & Mishra, 2008, p. 17–18).

In short, TPACK is a rich under-standing of how teaching and learning within a specific content area occur and change as a result of authentic, meaning-ful application of appropriate technolo-gies. “A teacher capable of negotiating these relationships represents a form of expertise different from, and greater than, the knowledge of a disciplinary expert (say a mathematician or histo-rian), a technology expert (a computer scientist), and a pedagogical expert (an experienced educator)” (Koehler, Mish-ra, & Polly, 2008, p. 1). Such an expert understands how technology influences decisions about content and pedagogy while also recognizing that content and pedagogy influence decisions about and uses of technology. As teachers think

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Volume 26 Number 4 | Journal of Digital Learning in Teacher Education | 133

about teaching specific concepts, they must concurrently be thinking about how and if technology can be used to make the concept more accessible and understandable to their students. This type of knowledge domain requires deep content knowledge, fluid pedagogical knowledge, and knowledge not only of technology tools, but knowledge about how to teach with these tools.

TPACK and MathematicsOne area that has seen dramatic growth in the influence and applications of technology on the development of con-tent and the evolution of instruction is mathematics. Math continues to evolve as a body of knowledge as technology1 influences what we know, how we know it, what we teach, and how we teach it. Technology has had considerable impact on the development and expansion of new and existing mathematical concepts and applications in the past few decades. For example, technology has allowed us to apply computer-like algorithms to create, analyze, and recursively define fractals, fragmented geometric shapes, objects, or quantities that are reduced-size copies (or self-similar structures) of the whole. Fractals have emerged as especially useful applications in defining and measuring geographic and meteoric features and phenomenon. Similarly, technology has influenced content development and exploration in areas such as statistics, combinatorics, algebra, probability, geometry, and matrices by providing novice and expert mathemati-cians increased access, understanding, and application of advanced mathemati-cal concepts through concrete model-ing, iterative applications, and recursive functioning (Grandgenett, 2008).

Technology has also had considerable impact on how we think about teach-ing mathematics, especially at the K–12 level. The National Council of Teachers of Mathematics (NCTM), in its Principles

1 Here and throughout this discussion, the term technology refers to contemporary instructional and learning technologies, typically avail-able in desktop or handheld form, that engage teachers and students in the teaching/learning process by promoting “…interactivity, multi-modality, various new forms of communication, access to expertise, new varieties of resources, opportunities for stimulation, enhanced productivity, and so on” (Herman, 1994, p. 133).

and Standards for School Mathematics(National Council of Teachers of Math-ematics, 2000), states:

Electronic technologies—calcula-tors and computers—are essential tools for teaching, learning, and doing mathematics. They furnish visual images of mathematical ideas, they facilitate organiz-ing and analyzing data, and they compute efficiently and accu-rately.… When technological tools are available, students can focus on decision making, reflection, reasoning, and problem solving. (p. 24).

The Association of Mathematics Teacher Educators (AMTE) (2006), in a recent position paper, agreed with the NCTM and further stated that “technology has become an essential tool for doing math-ematics in today’s world, and thus … it is essential for the teaching and learning of mathematics” (p. 1).

Properly implemented, technology changes how mathematics is taught by allowing teachers and students to focus on deep conceptual understanding over rote procedural skills through prob-lem solving, reasoning, and decision making. There are many ways in which technology can be used to foster this type of mathematical thinking. For example, dynamic software environ-ments, such as Geometer’s Sketchpad, Cabri, Fathom, or Tinkerplots, make

the exploration of core mathematical concepts tangible and interactive for students. These type of environments allow students to “…build and inves-tigate mathematical models, objects, figures, diagrams, and graphs,” (Key Curriculum Press, 2008, para. 1) in ways that bridge the gap between con-crete and abstract. Handheld graphing devices allow students, through explo-rations and applications, to develop a deeper understanding of mathematical concepts and use higher-level ap-proaches to solve mathematical prob-lems. Handhelds also promote assimi-lation between mathematical concepts and their multiple representations (e.g., functions and their graphical, tabular, and symbolic representations). Wireless network technologies, such as the TI Navigator, promote improved student engagement, understanding, and performance by allowing for real-time tracking of student progress, col-laborative lesson engagement, and in-stant feedback. Finally, virtual learning environments actively involve students in interactive mathematics instruction. Students are able to manipulate “physi-cal” objects to visualize relationships and applications, form and test conjec-tures, and connect abstract concepts to concrete representations.

As researchers continue to pay atten-tion to pedagogically appropriate uses of technology in professional devel-opment and classroom settings (e.g.,

Figure 1. TPACK framework (source: www.tpck.org).

Pedagogical Content Knowledge Pedagogical

Knowledge

Technological Pedagogical Knowledge

Technological Pedagogical Content Knowledge

Content Knowledge

Technological Content Knowledge

Technological Knowledge

C P

T



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Means, 1994), theoretical development of content-specific technology-related knowledge domains continue to emerge. This article adds to the development of mathematical TPACK by identifying key components of technology use in the mathematics classroom, demonstrat-ing robust TPACK through a classroom vignette, and addressing instructional implications of technology use through the development of questions to guide the technology-using mathematics teacher. The skills and knowledge related to tech-nology use in the mathematics classroom are very similar to those referred to in other knowledge domains, and yet they are vastly different because of their reli-ance on technology as an instructional and educational tool. Technology has the unique characteristic of being an instruc-tional tool that requires a specific set of operational skills while also necessitating a diverse array of instructional strategies and procedures.

Components of Mathematical TPACKThe knowledge needed to effectively employ technology as part of mathe-matics instruction includes technology-specific management, instructional, and pedagogical knowledge; increased mathematics subject-matter knowl-edge; and knowledge of when and how best to use technology to support mathematics instruction. TPACK in mathematics goes beyond knowledge of learning a technology tool and its op-eration, per se, to the dimension of how to operate a piece of technology to im-prove mathematics teaching and learn-ing. Although this knowledge includes learning the basic operational skills, it embodies the aspects of technology most relevant to its capacity for use in instruction to improve teaching and learning. Nowhere are these intricacies of technology’s effect on content and instruction more varied and applicable than in the mathematics classroom,

where technology has the potential to change not only what we teach but how we teach it.

This article argues that TPACK in mathematics can be characterized by four central components (see Figure 2). The first component, a teacher’s concep-tion and use of technology, relates tech-nology to pedagogical content knowl-edge by focusing on how the teacher can use technology to make the subject mat-ter more comprehensible and accessible to students. The next two components of TPACK include elements associated with general pedagogical knowledge. These components encompass the general principles of instruction, orga-nization, and classroom management specific to the application of technol-ogy in the mathematics classroom. The final component of TPACK relies on a teachers’ subject-matter knowledge and deals with the increased responsibility teachers have to understand their con-tent areas with both breadth and depth as a result of using technology as part of their instruction.

It should be noted that the number of TPACK components should not be thought of as a one-to-one correspon-dence with the subdomains of knowledge (i.e., PCK, TCK, and TPK) presented in the TPACK model. Rather, it is the result of careful analysis of seminal factors involved in technology-related pedagogy and subject matter specific to the content area under discussion. Consequently, this mathematical TPACK model posits one component related to each of PCK and TCK and two components related to TPK. This is not to say that TPK carries a greater weight or more importance in the devel-opment of TPACK in mathematics, but the issues related to instruction and manage-ment warranted their own consideration. Other content areas or conceptualizations of TPACK may include greater or fewer components, depending on the roles and interactions of technology, content, and pedagogy in the development of TPACK. As mentioned previously, the way that technology influences both pedagogy and content in mathematics is vastly different than the ways it may influence other con-tent areas, so core components of TPACK

Figure 2: Four central components of mathematics-related TPACK and their derivation from pedagogical content knowledge (PCK), technological pedagogical knowledge (TPK), and technological content knowledge (TCK).

C P

T

TPACK

TCK TPK

PCK

Conception & Use of Technology

Technology-Based Classroom Management

Technology-Based Mathematics Instruction

Depth & Breadth of Content



TPACK in the Math Classroom

may differ between mathematics and other content areas.

Conception and Use of TechnologyThe first component of TPACK includes a teacher’s overarching conception of the use of technology in support of teaching and learning mathematics. It includes what the teacher believes about mathematics as a field, how he or she feels mathematics can best be addressed through the use of technology, and what is important for students to learn about mathematics through the use of technolo-gy. This component serves as the basis for the instructional and curricular decisions teachers make in rendering the subject matter more accessible to students.

The teacher must decide how to best use technology (if at all) to address the needs of the students, the content, and instruction and then decide which technology best accomplishes all these goals. Is exploratory data analysis (EDA) best taught through the use of spread-sheets, graphing calculators, or one of the many statistical programs available for all levels of students? There is no single correct answer to this question; the answer will depend largely on the skills of the teacher with various types of technology; the specific topic he or she is investigating within EDA; and the students’ skills, needs, motivations, and prior understanding.

Most important, this component includes the knowledge of how to use technology in pedagogically appro-priate ways that support instruction authentically rather than as a side-show tool. Although researchers are able to identify what is not “pedagogically appropriate,” such as using technology for drill-and-skill or rote computation, they are less willing to specify exact elements of what appropriateness looks like. Because so much of appropriate technology use depends on the specific needs and nuances of each student, class, and teacher, pedagogically ap-propriate use of technology may vary from classroom to classroom. However, most researchers agree that, in general, pedagogically appropriate use implies a seamless integration of technology that

promotes inquiry, reasoning, contex-tualized learning, and sense-making (Guerrero, 2005).

In mathematics, pedagogically ap-propriate uses of technology encompass the ways technology is used to represent and formulate mathematics so that it is understandable to students through the most useful representations, demon-strations, examples, and applications. For example, when teaching fractals, how should a teacher use technology to represent them and most effectively demonstrate their applications? Would an online virtual manipulative demon-strate their creation most efficiently? Would a technical computing environ-ment make investigation into their iterative nature more accessible? Would dynamic geometry software demonstrate their applications most effectively? Of course, the answer depends on the exact nature of the content being covered, the instructional objectives, which tools are available to the teacher, and the tools with which he or she is most skilled.

As there is no single technologi-cal tool that is best for all instructional purposes, teachers must also be aware of the growing variety of tools available for their mathematics instruction. Within the mathematics classroom, this includes a thorough understanding of the uses and applications of, among other things, graphing calculators and their various programs and applications; data collec-tion devices such as calculator-based rangers (CRBs), computer-based labora-tories (CBLs), and other related scientific probes; spreadsheets; statistical software programs; dynamic geometry software programs; mathematical modeling and technical computing environments; online dynamic manipulatives; and the burgeoning use of networking tools such as TI Navigator. Though cost, time, and other factors limit teachers’ instructional decisions, especially where technology is concerned, technology-using teachers must have a solid foundation for the deci-sion of choosing one technology tool over another. Teachers must possess not only the knowledge of how to use the various features of each of these technologies, but also a thorough conceptualization of

when and how to use them as instruc-tional tools.

Technology-Based Mathematics Instruction The second component of TPACK includes teachers’ knowledge of and ability to maneuver through various instructional issues specifically related to the use of technology in support of mathematics teaching and learning. From this point of view, teachers need to understand that technology should be viewed as one instructional tool among many. It does not replace the teacher or any type of instruction, but should be included as part of a teacher’s instruc-tional repertoire.

Technology’s success as a learning and instructional tool depends upon it being integrated into a meaningful curricular and instructional framework, and it should be used only when it is the most appropriate means of reaching an instructional goal (Sandholtz, Ringstaff, & Dwyer, 1997). Is an investigation of symmetry better suited to miras, geo-boards, or dynamic geometry software? Then, once technology has been selected as the tool of choice, the teacher must decide which of various types of tech-nology are best suited to the learning objectives and content of a given lesson. Is an investigation of linear functions better suited to graphing calculators, spreadsheets, virtual manipulatives, or dynamic geometry software?

Also included in this component is the teacher’s ability to orchestrate the classroom environment in light of new demands and opportunities created by the use of technology. With the flourish-ing number of mobile and networking technologies, such as SmartBoards, inter-active slates, and the TI Navigator system, teachers must have the ability to manage collaborative inquiry and share control of the technology with students and among students (Goss, Renshaw, Galbraith, & Geiger, 2000). As such, technology-using mathematics teachers need to be aware of and comfortable with a didactic shift in attention from them to the topic the class is exploring, and in their role as part ofteacher-directed instruction versus




student-centered collaboration. They also need to recognize that technology may disrupt their instructional plans by uncovering insights into new and unexpected areas, and teachers should be comfortable with adapting to and making spontaneous changes in instruc-tion. Now, more than ever, teachers need to be comfortable with and knowledge-able enough to go with that “teachable moment.” By the same token, though, teachers should be aware of their re-sponsibility to set boundaries on how far students can and should go in their investigations and individual work.

Finally, this instructional component of TPACK includes the ability to adjust the use of technology to serve the needs of a diverse array of students in terms of mathematical ability, affect, and inter-est. Just as students may lose interest in an assignment when it is too easy or too difficult, they may lose interest in using technology when its use does not take their personal needs into account (Sandholtz, Ringstaff, & Dwyer, 1997). Technology has long been used as a remediation tool, but some technology tools, such as the TI Navigator system, may actually make modified instruction to serve individual needs even easier by allowing the teacher to send, via a class-room network, different sets of students different sets of problems, instructions, or tasks.

Management The third component of TPACK in mathematics covers management issues specifically related to teaching and learn-ing with technology. The use of technol-ogy in instruction introduces a number of management variables and issues that teachers seldom encounter when their instruction does not use technology. Included here is a teacher’s understand-ing of how to handle students’ attitudes toward technology and their behavior as a result of using technology. How does one deal with issues such as students sending games and messages via graph-ing calculators, abusing data gathering probes, or using computers as physical shields to hide off-task behavior? On the other hand, teachers need to understand

that some technologies may actually make management of instruction and behavior easier by providing constant access to student activity, progress, and understanding. The TI Navigator system allows teachers to grab screenshots of student work on the calculator at any point in time and provides teachers with instantaneous, formative assessment capabilities at any point in the lesson.

Management also encompasses teachers’ understanding of how to deal with the physical environment (e.g., lighting, glare, setup of equipment, physical layout of room) and techni-cal problems (e.g., broken hard drives, jammed printers, network problems, software restrictions, worn-out batteries) that arise as a result of using technol-ogy. Although ability to deal with such logistical elements of technology use im-proves over time, early on there is often a steep learning curve associated with managing all the physical and technical aspects of various technology tools in the mathematics classroom.

A final element of the management component of TPACK is a teacher’s ability to maintain student engagement once the novelty effect has worn off. The use of technology has been shown to have positive effects on student attitudes, on-task behavior, initiative, engagement, and experimentation, but when used too often, too infrequently, or inap-propriately, it can also result in student frustration, boredom, distraction, and unwillingness to transition to other ac-tivities (Sandholtz, Ringstaff, & Dwyer, 1997). As with any instructional tool, teachers need to know when and how to use technology to provide students with the most authentic learning environ-ment possible. Using technology with every activity and for every instructional purpose is just as futile as using direct instruction for every topic and lesson.

Depth and Breadth of Mathematics ContentThe fourth component of TPACK takes into account the increased responsibil-ity teachers have to understand their mathematics in breadth and depth. Plac-ing technology in the hands of students

gives them the power to explore math to a depth that may be unfamiliar to the teacher (Goss et al., 2000). As a result, teachers need to be confident in their ability to handle students’ investigations and inquiries. As with instructional flexibility, depth in content knowledge provides teachers with the ability and flexibility to explore, emphasize, or de-emphasize various mathematical topics that may arise in the course of instruc-tion and investigation. When a student, using a graphing calculator, discovers an interesting fact about the slope of a tangent line while graphing quadratics, the teacher must decide if the find-ings are relevant and worth pursuing or tangential and best left alone. This requires content knowledge of not only functions and derivatives, but also a broader understanding of mathematics, the mathematics curricula, and where/how derivatives fit into the scope and sequence of both.

Along with having an extremely strong knowledge base in their subject matter, teachers must also possess a willingness to acknowledge their own subject-matter shortcomings. As a result of the depth and breadth of content that can be explored through technol-ogy, teachers need to understand that students may encounter topics and ideas that teachers may be unprepared to manage or address. In lieu of un-derstanding every possible avenue a student’s investigation and insight may take, the teacher needs to be able to acknowledge that they are unsure of a student’s discovery, comments, or questions and must be willing to invest the time and energy to investigate these various content trails on their own.

A Portrait of TPACK in ActionPerhaps one of the best ways to grasp the complexity of TPACK in action is to examine each of the four components in the context of one teacher’s technology use. Barbara, a secondary mathematics teacher at a rural high school in central California, has been teaching for 18 years and recently moved from the middle grades to her new position at her district’s only high school. She has long been a

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proponent of technology in support of mathematics teaching and learning and spearheaded a controversial effort to require graphing calculators while at the middle school. She was an active partici-pant in a long-term technology profes-sional development program within the state and continues to be an ardent proponent of technology at all grade levels in her district. Barbara’s technol-ogy use is, for lack of a better word, fluid. She demonstrates robust mathematical TPACK through her conceptions and use of technology, her technology-based math instruction and management skills, and her depth and breadth of mathemati-cal content knowledge.

Conception and Use of TechnologyBarbara is attracted to technology as an instructional tool because of its poten-tial to improve her students’ learning and depth of understanding. She uses technology because she feels it allows students to learn about “current math” and to learn math with more depth. She feels that technology can be applied in ways that make learning math more meaningful for the students and that technology allows her and her students to do things they would not be able to do otherwise. Technology enables her students to “see the math,” make con-nections, and go more in depth with in-creased understanding. Barbara reflects her beliefs in the benefits of technology through the decisions she makes about using and implementing technology. In her own words, she sees the “big pic-ture” and believes technology provides students with useful experiences that connect mathematics to their daily lives, and prepares them for a technologically savvy “real world.”

Technology-Based Mathematics InstructionMost of the decisions Barbara makes regarding technology use are aimed at making technology a natural part of the learning process rather than an object of study in and of itself. “I am just continu-ally looking for how to make it seamless between teaching the math,” she ex-plains. Despite her obvious enthusiasm

for and commitment to technology as an instructional tool, Barbara believes that students should be taught to think about technology as one of many tools in their mathematical repertoire. “It’s a tool like anything else, like spell check or like pencils or using compasses,” she says. For Barbara, technology should be used with, rather than blindly replace, other tools, such as mental math and manual computation with paper and pencil.

When she uses technology to teach a topic, it is clear that Barbara has care-fully planned the lesson and knows where she wants it to end up. However, she is open to using student input to guide the nuances of how the lesson will get there. Barbara often uses question-ing to guide students in a whole-class discovery-based discussion focusing on real-world theme problems that often span the course of several days. Barbara believes that students learn best when they see the connections within math-ematics and the real world. For Barbara, technology is crucial to helping students make these connections by providing them with hands-on experiences and instant visuals, reinforcing concept links, and connecting math to real-world applications. In her attempts to make technology use seamless, Barbara plans her lessons so that students learn the technology as they learn the mathemat-ics. As she introduces new material, she reinforces old technology-related skills and integrates new skills.

Depth and Breadth of ContentIn a typical lesson, Barbara gives students a focus problem at the beginning of class, and, though the use of various question-ing and discussion techniques, she gives students instructions for setting up their calculators and guides them through the exploration of a mathematical topic. These lessons are very interactive and involve a lot of give and take between Barbara and her students. These “theme problems” often run the course of several days. For example, students were investi-gating the rate of change in millimeters of the lean on the Tower of Pisa over the last 100 years. Although the problem started out as a theme problem that the class

would pursue through two days of data analysis, it expanded into a problem that had students researching Tower of Pisa facts at home, arguing about an outlying point created through human interfer-ence versus natural causes, and wanting to explore center of gravity so they could determine when the tower would fall if its lean continued increasing at the current rate. This led to mini research projects, investigation of statistical outliers and their meaning, and construction of mod-els to replicate “lean and fall” scenarios.

In follow-up interviews, Barbara men-tioned that the problem had “morphed out of control” in several different direc-tions than the one she intended, but she was going with what the students wanted to do because they were engaged, exploring some really rich content, and getting at the heart of some really teach-able moments. Students were, of their own volition, asking about and investi-gating rich crosscurricular content areas that demonstrated not only true interest, but depth of understanding. She be-lieved that engaging them through their own questions was well worth the risk of not necessarily knowing the outcome. Though she did not know the answer to every question the students asked, Barbara demonstrated flexibility in her willingness to explore content areas beyond her immediate grasp.

ManagementBecause of her extensive background with all types of technology, Barbara is very comfortable and proficient with technology, likes to try new things, and is willing to take the risk of experiment-ing in front of her students. She has become an expert at troubleshooting graphing calculators in a multi-platform setting. Although she now uses TI graphing calculators in her teaching, some of her students still own and use Casios that were required when they were in middle school. Barbara is able to move effortlessly from helping a TI student to helping a Casio student. Because she gives instructions only on the TI, Barbara often pairs her few Casio students with one another and frequently checks in with them to make




sure they are able to follow along. By her own account, part of her success at troubleshooting comes from her strict protocol for handling technology. She tells students not to touch the calculator screens, not to use anything other than their fingers to push the buttons, and to place their calculators on their desks at all times. “I don’t know how much difference it makes,” she admits. “It’s just that I am more comfortable staying fo-cused on math if I can walk around the room and see what they are doing, and I can troubleshoot faster.”

Barbara exemplifies a teacher with comprehensive TPACK through her balanced use of technology as one tool within her instructional repertoire and her grounded beliefs that it is her responsibility to help prepare students for the tech-savvy real world. She firmly believes in the benefits of technology for making content accessible to her students but uses it within a meaningful curricular framework. Her teaching emphasizes col-laborative inquiry through student-cen-tered discussions and activities and often focuses on larger theme problems that run the course of several days. She has a solid background in mathematics and, though she clearly has instructional and content goals in mind, she is willing to explore unfamiliar areas if that is where student inquiries take a lesson.

DiscussionWhen choosing to use technology as part of their instructional repertoire, teachers must understand elements and implications of technology use related to instruction, management, content, pedagogy, and technology itself. Though Barbara’s illustration provides an example of one teacher’s use and conception of technology, the journey toward becoming such an authentic technology-using teacher takes time, energy, and commitment. Whether a novice technology user or a more experienced one such as Barbara, technology-using teachers are con-tinually changing and growing in their conceptions and use of technology.

Questions, such as those provided in Table 1, provide a springboard for discussion and reflection centered on each of the four components of TPACK. These questions prompt theoretical and practical deliberation by both experi-enced and novice teachers, individuals and groups engaged in a technology-based change process, and teacher educators. Teachers will address various components of TPACK in different ways and must rely on their own expertise to begin thinking about some of the theoretical aspects of the application of technology in support of mathematics instruction.

ConclusionDevelopment and understanding of TPACK, especially as it relates to specific content areas, is imperative because of the importance of technology’s ap-propriate use in educational settings. If technology is to influence teachers’ practices in reform-oriented ways and improve students’ learning by hav-ing a positive impact on engagement, achievement, and confidence, it must be successfully integrated into instruction in effective, authentic, and nonroutine ways. Ensuring technology’s proper use in educational settings requires the development and understanding of the characteristics of teachers’ technological pedagogical content knowledge base.

This article has attempted to ad-dress some of the central components of practical TPACK for the mathematics classroom. These include, but are not limited to, conception and use of tech-nology; technology-based mathematics instruction and management; and depth and breadth of mathematics content. Al-though implicit in this knowledge base is facility with basic operational skills for various types of technology, TPACK most notably embodies the aspects of technology most relevant to its ability to be used as part of a teacher’s instruc-tional repertoire to improve teaching

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Table 1: Questions to Guide the Development and Use of Technology by Analyzing Each Component of TPACK

Component Question

Instruction Is technology the most appropriate instructional tool for teaching and learning this topic?How will technology affect the collaborative nature of my classroom?Will I be able to adapt my instruction based on student feedback, progress, and/or inquiry with technology?How will I adjust my instruction and the use of technology to meet individual student needs?

Management How will I manage the physical logistics of technology? Where will we use technology? How many students per computer/calculator? Can I troubleshoot technical and/or application problems?How do I manage student progress and behavior?How do I encourage and maintain student engagement with technology-based lessons?

Depth and Breadth of Content Do I have the mathematical knowledge to handle student inquiries that may take us beyond the intent of this lesson?Am I willing to acknowledge my own content-related shortcomings and invest the time and energy to investigate student-generated “content trails”?

Conception and Use of Technology Is this topic best addressed through the use of technology? If so, how?What should students learn about this topic through the use of technology?How does technology improve teaching and learning of this topic?Is the use of technology in this lesson pedagogically appropriate?Do I have the skills to operate, navigate, and apply the various features of mathematics-related technology tools?Is technology fully integrated into this lesson or an add-on?Which technology will best support teaching and learning of a specific topic?



and learning. The working technology knowledge of a mathematics teacher using graphing calculators, computer software programs, and computer-based laboratories to deeply explore a mathe-matical topic is vastly different than that of an English teacher using the Internet and software programs to investigate and prepare literary documents. Each content area has specific instructional goals and needs that technology can address in a variety of ways. TPACK em-bodies a teacher’s ability to distinguish between the types of technology that are most suited to their content area and make decisions regarding its appropriate application.

Author NotesShannon Guerrero, PhD, is an assistant professor of mathematics education in the Department of Mathematics and Statistics at Northern Arizona University in Flagstaff, Arizona. Her current instructional responsibilities include both un-dergraduate and graduate courses in secondary mathematics methods, mathematics for elemen-tary teachers, and mathematics educational technology. Her research and professional interests include the use of Web 2.0 technology in support of mathematics teaching and learning, technol-ogy-related professional development, classroom practices and the impetus for teacher change, and inservice/preservice mathematics teacher educa-tion. Correspondence regarding this article should be addressed to Shannon Guerrero, Northern Arizona University, Department of Mathematics & Statistics, PO Box 5717, Flagstaff, AZ 86004. E-mail: [email protected]

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